Intuition about unit vectors in $n$-dimensional space

Question

Given two $n$-dimensional vectors on the unit sphere, $\mathbf{x}, \mathbf{y} \in \mathbb{S}^{n-1}$, is it true that if $\mathbf{x}^{\mathbf{T}}\mathbf{y}=-1$ then $\mathbf{x}=-\mathbf{y}$ ? What about $\mathbf{x}^{\mathbf{T}}\mathbf{y}=1 \Rightarrow \mathbf{x}=\mathbf{y}$ ? It seems difficult to see what is going on when the dimension is arbitrary.

Answer 1

$$ \frac{1}{2} (x-y)^T(x-y) = \frac{1}{2} (x^T x - y^T x - x^T y + y^T y) = 1-x^Ty. $$ The left-hand side is zero if and only if $x=y$, which gives you one answer. You can find the other one by replacing $y$ by $-y$.

Answer 2

Hint: $|\langle x, y\rangle| = \Vert x \Vert \cdot \Vert y \Vert$, i.e. equality occurs in Cauchy-Schwarz, if and only if $x=ky$.